In this paper, we study condition $C'_{\wedge}$ which is a
projective tensor product analogue of condition $C'$. We show
that
the finite-dimensional OLLP operator spaces have condition
$C'_{\wedge}$ and $M_{n}$ $(n\gt 2)$ does not have that property.

Let $\ell\in\mathbb N$ and $\alpha\in (0,2\ell)$. In this article,
the authors establish
equivalent characterizations
of Besov-type spaces, Triebel-Lizorkin-type
spaces and Besov-Morrey spaces via the sequence
$\{f-B_{\ell,2^{-k}}f\}_{k}$ consisting of the difference between
$f$ and
the ball average $B_{\ell,2^{-k}}f$. These results give a way
to introduce Besov-type spaces,
Triebel-Lizorkin-type spaces and Besov-Morrey spaces with any
smoothness order
on metric measure spaces. As special cases, the authors obtain
a new characterization of Morrey-Sobolev spaces
and $Q_\alpha$ spaces with $\alpha\in(0,1)$, which are of independent
interest.

Let $1\leq p\lt \infty$, and let $G$ be a discrete group. We give
a sufficient and necessary condition
for weighted translation operators on the Lebesgue space $\ell^p(G)$
to be densely disjoint hypercyclic.
The characterization for the dual of a weighted translation to
be densely disjoint hypercyclic is also obtained.

In the Simpson moduli space $M$ of semi-stable sheaves with
Hilbert polynomial $dm-1$ on a projective plane we study the
closed subvariety $M'$ of sheaves that are not locally free on
their support. We show that for $d\ge 4$ it is a singular subvariety
of codimension $2$ in $M$. The blow up of $M$ along $M'$ is interpreted
as a (partial) modification of $M\setminus M'$ by line bundles
(on support).

In this paper we investigate
the endpoint regularity properties of the multisublinear
fractional maximal operators, which include the multisublinear
Hardy-Littlewood maximal operator. We obtain some new bounds
for the derivative of the one-dimensional multisublinear
fractional maximal operators acting on vector-valued function
$\vec{f}=(f_1,\dots,f_m)$ with all $f_j$ being $BV$-functions.

On a real hypersurface $M$ in a non-flat complex
space form there exist the Levi-Civita and the k-th generalized
Tanaka-Webster connections. The aim of the present paper is to
study three dimensional real hypersurfaces in non-flat complex
space forms, whose Lie derivative of the structure Jacobi operator
with respect to the Levi-Civita connections coincides with the
Lie derivative of it with respect to the k-th generalized Tanaka-Webster
connection. The Lie derivatives are considered in direction of
the structure vector field and in directions of any vecro field
orthogonal to the structure vector field.

Let $\mathbb{L}$ be a length function on a group $G$, and let $M_\mathbb{L}$
denote the
operator of pointwise multiplication by $\mathbb{L}$ on $\lt(G)$.
Following Connes,
$M_\mathbb{L}$ can be used as a ``Dirac'' operator for the reduced
group C*-algebra $C_r^*(G)$. It defines a
Lipschitz seminorm on $C_r^*(G)$, which defines a metric on the
state space of
$C_r^*(G)$. We show that
for any length function satisfying a strong form of polynomial
growth on a discrete group,
the topology from this metric
coincides with the
weak-$*$ topology (a key property for the
definition of a ``compact quantum metric
space''). In particular, this holds for all word-length functions
on finitely generated nilpotent-by-finite groups.

In this paper, using CalderÃ³n's
reproducing formula and almost orthogonality estimates, we
prove the lifting property and the embedding theorem of the Triebel-Lizorkin
and Besov spaces associated with Zygmund dilations.

This paper introduces a unified operator theory approach to the
abstract Plancherel (trace) formulas over
homogeneous spaces of compact groups. Let $G$ be a compact group
and $H$ be a closed subgroup of $G$.
Let $G/H$ be the left coset space of $H$ in $G$ and $\mu$ be
the normalized $G$-invariant measure on $G/H$ associated to the
Weil's formula.
Then, we present a generalized abstract notion of Plancherel
(trace) formula for the Hilbert space $L^2(G/H,\mu)$.

Let $X(n),$ for $n\in\mathbb{N},$ be the set of all subsets of a metric
space $(X,d)$ of cardinality at most $n.$ The set $X(n)$ equipped
with the Hausdorff metric is called a finite subset space. In
this paper we are concerned with the existence of Lipschitz retractions
$r\colon X(n)\to X(n-1)$ for $n\ge2.$ It is known that such retractions
do not exist if $X$ is the one-dimensional sphere. On the other
hand L. Kovalev has recently established their existence in case $X$
is a Hilbert space and he also posed a question as to whether
or not such Lipschitz retractions exist for $X$ being a Hadamard
space. In the present paper we answer this question in the positive.

We study some geometric properties related to the set $\Pi_X:=
\{
(x,x^*
)\in\mathsf{S}_X\times \mathsf{S}_{X^*}:x^*
(x
)=1
\}$ obtaining two characterizations of Hilbert spaces
in the category of Banach spaces. We also compute the distance
of a generic element $
(h,k
)\in H\oplus_2 H$ to $\Pi_H$ for $H$ a Hilbert space.

The
eigenvalue problem $-\Delta_p u-\Delta_q u=\lambda|u|^{q-2}u$
with $p\in(1,\infty)$, $q\in(2,\infty)$, $p\neq q$ subject to
the
corresponding homogeneous Neumann boundary condition is
investigated on a bounded open set with smooth boundary from
$\mathbb{R}^N$ with $N\geq 2$. A careful analysis of this problem leads
us to a complete description of the set of eigenvalues as being
a
precise interval $(\lambda_1, +\infty )$ plus an isolated point
$\lambda =0$. This comprehensive result is strongly related to
our
framework which is complementary to the well-known case $p=q\neq
2$ for which a full description of the set of eigenvalues is
still
unavailable.

The purpose of this paper is to characterize positive measure
$\mu$ on the unit disk such that the analytic
Morrey space $\mathcal{AL}_{p,\eta}$ is boundedly and compactly
embedded to the tent space
$\mathcal{T}_{q,1-\frac{q}{p}(1-\eta)}^{\infty}(\mu)$ for the
case $1\leq q\leq p\lt \infty$
respectively. As an application, these results are used to
establish the boundedness and compactness of integral operators
and multipliers between analytic Morrey spaces.

Let $\Gamma$ be a connection on a smooth manifold
$M$, in this paper we give some properties of $\Gamma$ by studying
the corresponding Lie algebras. In particular, we compute the
first Chevalley-Eilenberg cohomology space of the horizontal
vector fields Lie algebra on the tangent bundle of $M$, whose
the corresponding Lie derivative of $\Gamma$ is null, and of
the horizontal nullity curvature space.

In this paper, the authors characterize second-order Sobolev
spaces $W^{2,p}({\mathbb R}^n)$,
with $p\in [2,\infty)$ and $n\in\mathbb N$ or $p\in (1,2)$ and
$n\in\{1,2,3\}$, via the Lusin area
function and the Littlewood-Paley $g_\lambda^\ast$-function in
terms of ball means.

We study the existence of fixed points for contraction multivalued
mappings in modular metric spaces endowed with a graph. The
notion of a modular metric on an arbitrary set and the corresponding
modular spaces, generalizing classical modulars over linear spaces
like Orlicz spaces, were recently introduced. This paper can
be seen as a generalization of Nadler's and Edelstein's fixed
point theorems to modular metric spaces endowed with a graph.

The main aim of the present paper is to investigate various structural
properties
of hyperplanes of $c$, the Banach space of the convergent sequences.
In particular, we give an explicit formula for the projection
constants and we prove that an hyperplane of $c$ is isometric
to the whole space if and only if it is $1$-complemented. Moreover,
we obtain the classification
of those hyperplanes for which their duals are isometric to
$\ell_{1}$ and we give a complete description of the preduals
of $\ell_{1}$ under the assumption that the standard basis of
$\ell_{1}$
is weak$^{*}$-convergent.

One can easily show that any meromorphic function
on a complex closed Riemann surface can be represented as a
composition of a birational map of this surface to $\mathbb{CP}^2$ and
a projection of the image curve from an appropriate point
$p\in \mathbb{CP}^2$ to the pencil of lines through $p$. We introduce
a natural stratification of Hurwitz spaces according to the
minimal degree of a plane curve such that a given meromorphic
function can be represented in the above way and calculate the
dimensions of these strata. We observe that they are closely
related to a family of Severi varieties studied earlier by J. Harris,
Z. Ran and I. Tyomkin.

We recently introduced a weighted Banach algebra $\mathfrak{A}_G^n$ of
functions which are holomorphic on the unit disc $\mathbb{D}$, continuous
up to the boundary and of the class $C^{(n)}$ at all points where
the function $G$ does not vanish. Here, $G$ refers to a function
of the disc algebra without zeros on $\mathbb{D}$. Then we proved that
all closed ideals in $\mathfrak{A}_G^n$ with at most countable hull are
standard. In the present paper, on the assumption that $G$ is
an outer function in $C^{(n)}(\overline{\mathbb{D}})$ having infinite roots
in $\mathfrak{A}_G^n$ and countable zero set $h(G)$, we show that all the
closed ideals $I$ with hull containing $h(G)$ are standard.

In this paper we define $VMO_\mathcal{P}$ space associated with
a family $\mathcal{P}$ of parabolic sections and show that the
dual of $VMO_\mathcal{P}$ is the Hardy space $H^1_\mathcal{P}$.
As an application, we prove that almost everywhere convergence
of a bounded sequence in $H^1_\mathcal{P}$ implies weak* convergence.